A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces

doi:10.1007/s42967-023-00303-8

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (2): 1189-1216.doi: 10.1007/s42967-023-00303-8

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A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces

Young Kyu Lee, Shingyu Leung

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

Received:2022-10-28 Revised:2023-08-01 Accepted:2023-08-04 Online:2023-10-16 Published:2023-10-16
Contact: Shingyu Leung,E-mail:masyleung@ust.hk;Young Kyu Lee,E-mail:ykleeac@connect.ust.hk E-mail:masyleung@ust.hk;ykleeac@connect.ust.hk
Supported by:
The work of Leung was supported in part by the Hong Kong RGC 16302223.

Abstract

Abstract: We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces. The approach follows an embedding approach for solving the surface eikonal equation. We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood. Our proposed algorithm is easy to implement and efficient. We will give some two- and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

Key words: Laplace-Beltrami operator, Level set method, Implicit representation, Eigenvalues, Numerical PDEs

Young Kyu Lee, Shingyu Leung. A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 1189-1216.

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A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces

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